• Korean Journal of Mathematics
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• v.20 no.4
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• pp.403-414
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• 2012

A voltage assignment on a graph was used to enumerate all possible 2-cell embeddings of a graph onto surfaces. The boundary of the surface which is obtained from 0 voltage on every edges of a very special diagram of a complete bipartite graph $K_{m,n}$ is surprisingly the ($m,n$) torus link. In the present article, we prove that every link is the boundary of a complete bipartite multi-graph $K_{m,n}$ for which voltage assignments are either -1 or 1 and that every link is the boundary of a complete bipartite graph $K_{2,n}$ for which voltage assignments are either -1, 0 or 1 where edges in the diagram of graphs may be linked but not knotted.

• Kyungpook Mathematical Journal
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• v.57 no.2
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• pp.301-307
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• 2017

In this paper, we solve the following four graph equations $L^k(G)=H{\oplus}J$; $M(G)=H{\oplus}J$; ${\bar{L^k(G)}}=H{\oplus}J$ and ${\bar{M(G)}}=H{\oplus}J$, where J is $nK_2$ for $n{\geq}1$. Here, the equality symbol = means the isomorphism between the corresponding graphs. In particular, we shall obtain all pairs of graphs (G, H), which satisfy the above mentioned equations, upto isomorphism.

• Kyungpook Mathematical Journal
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• v.56 no.3
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• pp.647-655
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• 2016

Let H be a graph, and $k{\geq}{\chi}(H)$ an integer. We say that H has a cyclic Gray code of k-colourings if and only if it is possible to list all its k-colourings in such a way that consecutive colourings, including the last and the first, agree on all vertices of H except one. The Gray code number of H is the least integer $k_0(H)$ such that H has a cyclic Gray code of its k-colourings for all $k{\geq}k_0(H)$. For complete bipartite graphs, we prove that $k_0(K_{\ell},r)=3$ when both ${\ell}$ and r are odd, and $k_0(K_{\ell},r)=4$ otherwise.

• Journal of KIISE:Computer Systems and Theory
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• v.31 no.1_2
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• pp.19-26
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• 2004

In this paper, we investigate the longest fault-free paths joining every pair of vertices in a double loop network with faulty vertices and/or edges, and show that a bipartite double loop network G(mn；1, m) is strongly hamiltonian-laceable when the number of faulty elements is two or less. G(mn；1, m) is bipartite if and only if m is odd and n is even.

• Honam Mathematical Journal
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• v.41 no.4
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• pp.863-870
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• 2019

In this paper, we obtain the enumeration results on the number of linear extensions of diamond posets. We find the recurrence relations and exponential generating functions for the number of linear extensions of diamond posets. We also get some results for the volume of graph polytope associated with bipartite graphs which are underlying graphs of diamond posets.

• Journal of applied mathematics & informatics
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• v.22 no.1_2
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• pp.21-38
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• 2006

Programmed grammars, one of the most important and well investigated classes of grammars with context-free rules and a mechanism controlling the application of the rules, can be described by graphs. We investigate whether or not the restriction to special classes of graphs restricts the generative power of programmed grammars with erasing rules and without appearance checking, too. We obtain that Eulerian, Hamiltonian, planar and bipartite graphs and regular graphs of degree at least three are pr-universal in that sense that any language which can be generated by programmed grammars (with erasing rules and without appearance checking) can be obtained by programmed grammars where the underlying graph belongs to the given special class of graphs, whereas complete graphs, regular graphs of degree 2 and backbone graphs lead to proper subfamilies of the family of programmed languages.

• Bulletin of the Korean Mathematical Society
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• v.59 no.3
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• pp.685-695
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• 2022

The minimum number of complete bipartite subgraphs needed to partition the edges of a graph G is denoted by b(G). A known lower bound on b(G) states that b(G) ≥ max{p(G), q(G)}, where p(G) and q(G) are the numbers of positive and negative eigenvalues of the adjacency matrix of G, respectively. When equality is attained, G is said to be eigensharp and when b(G) = max{p(G), q(G)} + 1, G is called an almost eigensharp graph. In this paper, we investigate the eigensharpness and almost eigensharpness of lexicographic products of some graphs.

• Kyungpook Mathematical Journal
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• v.63 no.3
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• pp.355-372
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• 2023

We make advances towards a structural characterisation of the signed graphs H for which the list switch H-colouring problem List-S-Hom(H) can be solved in polynomial time. We conjecture two different characterisations, the second refining the first, in the case that the graph H can be switched to a graph in which every negative edge is also positive. Using a recent proof of the first characterisations for reflexive signed graphs, by Bok et. al., we prove the second characterisation for reflexive signed graphs. We also provide several tools for reducing the problem to the bipartite case, and prove a full complexity dichotomy for a related problem.

• Kyungpook Mathematical Journal
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• v.58 no.4
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• pp.747-759
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• 2018

A graph G = (V (G), E(G) of order n = |V (G)| and size m = |E(G)| is said to be odd harmonious if there exists an injection $f:V(G){\rightarrow}\{0,\;1,\;2,\;{\ldots},\;2m-1\}$ such that the induced function $f^*:E(G){\rightarrow}\{1,\;3,\;5,\;{\ldots},\;2m-1\}$ defined by $f^*(uv)=f(u)+f(v)$ is bijection. While a bipartite graph G with partite sets A and B is said to be bigraceful if there exist a pair of injective functions $f_A:A{\rightarrow}\{0,\;1,\;{\ldots},\;m-1\}$ and $f_B:B{\rightarrow}\{0,\;1,\;{\ldots},\;m-1\}$ such that the induced labeling on the edges $f_{E(G)}:E(G){\rightarrow}\{0,\;1,\;{\ldots},\;m-1\}$ defined by $f_{E(G)}(uv)=f_A(u)-f_B(v)$ (with respect to the ordered partition (A, B)), is also injective. In this paper we prove that odd harmonious graphs and bigraceful graphs are equivalent. We also prove that the number of distinct odd harmonious labeled graphs on m edges is m! and the number of distinct strongly odd harmonious labeled graphs on m edges is [m/2]![m/2]!. We prove that the Cartesian product of strongly odd harmonious trees is strongly odd harmonious. We find some new disconnected odd harmonious graphs.

• Journal of Communications and Networks
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• v.13 no.5
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• pp.536-543
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• 2011

The heterogeneous postal delivery model assumes that each intermediate node in the multicasting tree incurs a constant switching time for each message that is sent. We have proposed a new model where we assume a more generalized switching time at intermediate nodes. In our model, a child node v of a parent u has a switching delay vector, where the ith element of the vector indicates the switching delay incurred by u for sending the message to v after sending the message to i-1 other children of u. Given a multicast tree and switching delay vectors at each non-root node 5 in the tree, we provide an O(n$^{\frac{5}{2}}$) optimal algorithm that will decide the order in which the internal (non-leaf) nodes have to send the multicast message to its children in order to minimize the maximum end-to-end delay due to multicasting. We also show an important lower bound result that optimal multicast switching delay problem is as hard as min-max matching problem on weighted bipartite graphs and hence O(n$^{\frac{5}{2}}$) running time is tight.